How many solutions does the system of equations have?
x=-4y+4 and 2x+8y=8
x = -4y + 4
Eq1: x + 4y = 4
Eq2: 2x+8y = 8
Divide Eq2 by 2:
x+4y = 4
The Eqs are identical and represent the same line. Therefore, we have an infinite number of solutions.
To determine how many solutions a system of equations has, we can solve them using various methods and check the resulting values of the variables.
Let's solve this system of equations using the substitution method. We have two equations:
Equation 1: x = -4y + 4
Equation 2: 2x + 8y = 8
First, let's solve Equation 1 for x in terms of y:
x = -4y + 4
Now, substitute this expression for x in Equation 2:
2(-4y + 4) + 8y = 8
Simplify the equation:
-8y + 8 + 8y = 8
0 = 0
When we simplify this equation, we find that 0 = 0, which means that the equation is always true. This indicates that the two equations represent the same line on the graph.
In this case, the system of equations is dependent, meaning that they have infinitely many solutions. Any value of y will satisfy both equations. The solution is therefore expressed in terms of one variable, y.
So, the system of equations has infinitely many solutions.